A cohomology Bott manifold which is not diffeomorphic to a Bott manifold 56

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold structureL:K→BStringobtained by the Lie-group framing, is the non-trivial element Ω^String₆ (pt) ∼= Ω^{f r}₆ (pt) ∼= Z/2, since its Arf-invariant (or Kervaire-Arf-invariant) is non-trivial. This was already shown in [KM63]. For a definition of the Arf-invariant we refer the reader to Chapter 6 in [L¨uc02]. We denote the non-trivial element in Ω^String₆ (pt) by [K, L].

Now we consider K×CP¹ together with the map L×pt:K×CP¹ → BString. By abuse of notation we denote this map by L, too. Since T(L×CP¹) ∼=T L⊕TCP¹ the normal bundle is given by ν(L×CP¹) ∼= ν(L)⊕ν(CP¹). Since the latter summand is stably trivialL induces a normal String structure onK×CP¹.

Lemma 4.11. Let pr₂:K×CP¹ →CP¹ be the projection upon the second factor. The element ω:= [K×CP¹, pr₂×L]∈Ω^String₈ (CP¹) is non-trivial. It is of finite order.

The construction which we use to prove the first part of the lemma is also known as codimension two Arf-invariant.

Proof. In Definition 3.17 we introduce the homomorphism t: Ω^String₈ (CP¹) → Ω^String₆ (pt)

[M, f ×α] 7→ [f⁻¹(f |∩pt)(f×α)|_f−1(f |∩^pt)].

By constructiont([K×CP¹, pr2×L]) = [K, L]6= 0. Thus, the preimage [K×CP¹, L×pr2] must be non-trivial in Ω^String₈ (CP¹).

The maptvanishes on im(Ω^String₈ (pt),→Ω^String₈ (CP¹)) by the exact sequence of Lemma 3.18. Thus, ω must be non-trivial under the projection to the reduced bordism group Ωe^String₈ (CP¹)∼=Z/2.

Next we show thatω is of finite order.

As in the proof of Theorem 4.2 we use the map pr₈:BString →BSOwhich induces a mapZ⊕Z/2∼= Ω^String₈ (pt)→Ω^SO₈ (pt) whose kernel is Z/2.

The Pontrjagin numbers are a complete set of invariants of Ω^SO₈ (pt). The first Pontrjagin class p₁(K ×CP¹) is an element in H⁴(K ×CP¹) which vanishes. Hence, the second Pontrjagin number p₍₂₎(K ×CP¹) := hp2(K ×CP¹),[K ×CP¹]i is, by the signature theorem, determined by the signature of K ×CP¹. But H⁴(K ×CP¹) = 0 implies that the signature vanishes. Thus, p₍₂₎(K×CP¹) vanishes, as well. This shows that the elementω ∈Ω^String₈ (CP¹) is contained in Ω^String₈ (pt)/Z⊕Ωe^String₈ (CP¹) which is the finite groupZ/2⊕Z/2.

Now we change [K×CP¹, pr₂×L] by surgery. By Proposition 4 of [Kre99] (which we cite in Proposition 3.8) we can turnpr₂×Linto a four-equivalence by surgery below the middle dimension. Since He_k(CP¹×BString) = 0 for 2 6=k ≤7 we obtain a manifold

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold Kp with

H_k(Kp)∼=

(Z fork= 0,2,6,8 0 else.

We denote this representative of [K×CP¹, pr₂×L]∈Ω^String₈ (CP¹) by (K_p, κ×νe_Kp).

Later on we want to be able to compare elements induced by T and the parametric connected sum - which we still need to explain - of T and K_p in Ω^String₈ (CP^∞). To be able to do this we need to understandK_p as an element in Ω^String₈ (CP^∞).

Lemma 4.12. The inclusion CP¹ →CP^∞ induces a monomorphism Ω^String₈ (CP¹)→Ω^String₈ (CP^∞).

In particular, [K_p, κ×νe_Kp]gives rise to a non-trivial element in Ωe^String₈ (CP^∞).

The strategy of the proof is the following. The Pontrjagin-Thom construction results in an isomorphism

Ω^String₈ (CP^∞)∼=π^st₈ (CP₊^∞∧M String).

Thus, we can apply the Adams spectral sequence to calculate Ω^String₈ (CP^∞). Then we can compare the Atiyah-Hirzebruch spectral sequences of Ω^String₈ (CP¹) and Ω^String₈ (CP^∞) since the inclusionCP¹→CP^∞induces a map on the respectiveE²-pages. By the calcu-lations of Ω^String₈ (CP¹) and Ω^String₈ (CP^∞) we also know the infinity pages. This enables us to deduce that the map

Ω^String₈ (CP¹)→Ω^String₈ (CP^∞) induced by the inclusionCP¹ →CP^∞ is injective.

Later on we calculate the E²-page of the Atiyah-Hirzebruch spectral sequence. There we see, that the only torsion that appears is two-primary. Consequently it is justified to restrict to the Adams spectral sequence at the prime two. Recall that theE₂-page of the Adams spectral sequence, converging to π^st_t−s(CP₊^∞∧M String), at the prime two, has entries

E₂^s,t= Ext^s,t_A (H^∗(CP₊^∞∧M String; Z/2),Z/2).

To calculate theE₂-page fort−s≤9 we use the method of minimal resolutions as intro-duced in [Sto85]. We calculate the necessary data for the method of minimal resolutions, i.e. the Steenrod module structure of H^∗(CP^∞∧M String; Z/2) in the proof. For this one example we also calculate the resolution explicitly in Appendix B. We checked the result by a computer algorithm developed by Bruner (cf. [Bru93] and [Bru]).

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold Proof. To calculate the minimal resolution we needH^k(CP₊^∞∧M String; Z/2) fork≤10.

The ringH^∗(CP^∞, Z/2) is generated bya∈H²(CP^∞; Z/2).

The cohomologyH^∗(BString; Z/2) is determined in [Sto63]. By the Thom isomorphism the only non-vanishing cohomology groups of M String in degree less or equal ten are H^k(M String; Z/2)∼=Z/2 for k= 0,8. The generator in degree zero is the Thom class u, the one in degree eight is uw₈. Here w₈ denotes the pullback of the eighth universal Stiefel-Whitney class inH⁸(BO;Z/2) to H⁸(BString;Z/2). By Chapter 8 in [MS74] we know Sq⁸(u) =uw₈.

We consider the pullback of the classes u, uw₈, a, a², ... toH^∗(CP₊^∞∧M String), apply the K¨unneth theorem and obtain

i 0 2 4 6 8 10

Hⁱ(CP₊^∞∧M String; Z/2) Z/2 Z/2 Z/2 Z/2 Z/2² Z/2² generators u ua ua² ua³ uw₈, ua⁴ ua⁵, w₈a

.

The other groupsHⁱ(CP₊^∞∧M String; Z/2) vanish fori≤10. Now, a straight forward calculation shows that the only non-vanishing operations of Steenrod squaresSqⁱ in this range are:

Sq⁸u=uw8, Sq²ua=ua², Sq⁸ua=uw₈a, Sq⁴ua² =ua⁴, Sq²ua³ =ua⁴, Sq⁴ua³ =ua⁵.

From this data we calculate the minimal resolution in Appendix B and obtain the fol-lowing E₂-page. Again, we indicate the multiplicative structure on the E₂-page as in Example 6.19 of [Sto85].

0 2 4 6 8

0 2 4 6 8

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold The entries for t−s=9 correspond to the coefficients Ω^String₉ (pt). Consequently, they must survive to the E∞-page. Hence, there cannot be any differential that hits the column (t−s) = 8 and we obtain

Ω^String₈ (CP^∞)∼= Ω^String₈ (pt)⊕Z⊕Z/2.

Since CP¹∼=S² we see

Ω^String₈ (CP¹)∼= Ω^String₈ (pt)⊕Ω^String₈ (S², pt)∼= Ω^String₈ (pt)⊕Ω^String₆ (pt) is isomorphic to Ω^String₈ (pt)⊕Z/2.

Now we start the comparison of theE²-pages of the Atiyah-Hirzebruch spectral sequences converging to Ω^String₈ (CP¹) and Ω^String₈ (CP^∞). For this we use thatCP¹ ,→CP^∞induces an injective map on homology groups.

Consider the Atiyah-Hirzebruch spectral sequence withE²-page E_pq² (CP^∞)∼=H_p(CP^∞; Ω^String₈ (pt))

converging to Ω^String_p+q (CP^∞). Since we are only interested in p+q = 8 we only depict the seventh, eighth and ninth diagonal and the coefficients.

0 2 4 6 8

0 3 6

Z Z/2 Z/2 Z/24

Z/2 Z⊕Z/2

Z/24

Z/2 Z/2

Z/2

Z Z/2⁸

Z/2

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold SinceSq²:H⁶(CP^∞;Z/2)→H⁸(CP^∞;Z/2) is an isomorphism, the indicated differential is an isomorphism by Lemma 3.16 as well. Thus, on theE³-page, there only remain two entries containing a Z/2. Since Ω^String₈ (CP^∞)∼= Ω^String₈ (pt)⊕Z⊕Z/2 both entries must survive to theE^∞-page.

We now compare the E²-pages. Denote the entry of the Atiyah-Hirzebruch spectral sequence converging to Ω^String₈ (CP¹) byE_pq² (CP¹) =H_p(CP¹; Ω^String_q (pt)). The inclusion CP¹ ,→CP^∞ induces injective maps

H₀(CP¹; Ω^String₈ (pt))∼=E₀₈² (CP¹) → E₀₈² (CP^∞)∼=H₀(CP^∞; Ω^String₈ (pt)) and H₂(CP¹; Ω^String₆ (pt))∼=E₂₆² (CP¹) → E₂₆² (CP^∞)∼=H₂(CP^∞; Ω^String₆ (pt)).

Since all these entries survive to E^∞, this proves the Lemma.

Now we construct the parametric connected sumT#_CP¹K_p and show that it actually is a cohomology Bott manifold.

By Hurewicz’s Theorem all classes in H₂(K_p) ∼= Z are spherical. Thus, we can fix an embeddingi:S² ,→K_p such thati∗[S²] generatesH₂(K_p). SinceK_pis a String manifold ν(S² ,→ K_p) is stably trivial. The rank of ν(S² ,→K_p) is bigger than the dimension of the sphere, whence the normal bundle is actually trivial. Consequently, we obtain an embedding S²×D⁶,→Kp.

The same holds for T, where we take the embedding to be s₄ ◦s₃ ◦s₂:CP¹₁ → T as defined in Section 2.1. Thus, we can cut S² ×D⁶ out of T and K_p and identify the boundaries along the identity. We call this theparametric connected sum and denote it by F :=T#_CP¹K_p, where F stands forfake Bott manifold.

Note that we could also construct F using the embeddings given by the appropriate compositions of sections and inclusions of the fiber, e.g. s₄◦i₃.

Lemma 4.13. The parametric connected sum F is a cohomology Bott manifold.

Recall that a cohomology Bott manifold is defined to be a manifold which admits a polarisation mapg:H^∗(T)→H^∗(F). We construct such a map g in the proof.

Proof. First we prove that there exists a ring isomorphism g:H^∗(T)→H^∗(F).

LetC_K be the complementK_p−S²×D⁶ and letC_T be the complement T−(S²×D⁶).

By the Mayer-Vietoris sequence ofKp =S²×D⁶∪C_K we getH^j(C_K)∼=H^j(S²×D⁶) for all j.

Similarly, by the Mayer-Vietoris sequence of T =C_T ∪S²×D⁶ and F =C_T ∪C_K, we obtain isomorphisms H^k(T) → H^k(C_T) and H^k(F) → H^k(C_T) for k = 0,2,4. Since these are induced by the inclusions i:C_T → T and j: C_T → F they are natural with respect to the cup product.

Combining both isomorphisms we obtain isomorphisms φ_k:=i^∗◦(j^∗)⁻¹:H^k(T)→H^k(F)

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold fork≤4 such that, for allx, y∈H²(T),φ2(x)∪φ2(y) =φ4(x∪y).

Let D⁶ ⊂ D⁶ be a disk such that the closure of D⁶ is contained in the interior of D⁶. Thus, we can apply excision to S² ×D⁶ ⊂ S²×D⁶ ⊂ T and obtain an isomorphism H^k(T, S²×D⁶)∼=H^k(C_T, ∂C_T).

Similarly we can constructCe_K ⊂C_K such that we can apply excision toCe_K ⊂C_K⊂F, whenceH^k(F, C_K)∼=H^k(C_T, ∂C_T).

Furthermore, by the long exact sequence of the pairs (T, S²×D⁶), (F, C_K) and (C_T, ∂C_T), we obtain a commutative diagram, where, fork= 4 all maps are isomorphisms:

H^k(T) ^//H^k(C_T)^oo H^k(F)

H^k(T, S²×D⁶)^{_ _ _//}

OO

H^k(C_T, ∂C_T)

OO

H^k(F, C_K). oo_ _ _

OO

Fork= 4 the dotted arrows formφ₄. Composing the dashed arrows we also obtainφ₄ by commutativity. Fork= 8 the dashed arrows are also natural isomorphisms. We denote their composition by φ₈:H⁸(T) →H⁸(F). By naturalityφ₄(x)∪φ₄(y) =φ₈(x∪y) for all x, y∈H⁴(T). In particular, this determines the intersection form onF.

Since all odd cohomology groups of F vanish, it remains to construct an isomorphism φ₆:H⁶(T)→H⁶(F) such that together allφ_k, fork= 0,2,4,6,8, constitute an isomor-phism of rings.

Recall that x_m :=y_m−α_m is a basis for H²(T) by Lemma 2.3.

For 1≤i, j, l, m≤4,i < j andl < m we obtain

φ2(yi)∪φ2(yj)∪φ2(y_l)∪φ2(xm) =φ4(yi∪yj)∪φ4(y_l∪xm) =φ8(yi∪yj∪y_l∪xm).

An explicit calculation shows that y_i ∪y_j ∪y_l ∪x_m is a generator, and thereby that φ₈(y_i∪y_j ∪y_l∪x_m) is a generator, if and only if{i, j, l, m}={1,2,3,4} and zero else.

Let l /∈ {i, j, k}. By the above observation the products φ2(yi) ∪φ2(yj)∪φ2(y_k) for i < j < k, form the basis of H⁶(F) consisting of the Kronecker duals ofφ₂(x_l)∩[F].

Defineφ₆(y_i∪y_j∪y_k) :=φ₂(y_i)∪φ₂(y_j)∪φ₂(y_k) fori < j < k. Together theφ_kconstitute a ring isomorphismg:H^∗(T)→H^∗(F).

It remains to show thatgis a polarisation map, i.e. that it preserves the Stiefel-Whitney and Pontrjagin classes.

To prove that g preserves the characteristic classes in degree less or equal four, we use the inclusions i:C_T → T and j:C_T → F. By construction φ₂ and φ₄ are (j^∗)⁻¹◦i^∗. The tangent bundles of T and F both pull back to the tangent bundle of C_T under i andj, respectively. Thus, by naturality,φ₂ andφ₄respect the second and fourth Stiefel-Whitney class and the first Pontrjagin class.

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold The Euler class of F and T is even. Thus, the top Stiefel-Whitney class vanishes since it is just the mod two reduction of the Euler class. The signatures ofF andT agree and p₁(T) = 0 =p₁(F). Hence, their second Pontrjagin classes are preserved underg by the signature theorem.

Consequently, it remains to show that g(w₆(T)) = w₆(F). We use the Wu classes to check this.

Recall that the Wu classes vi(X) of an n−dimensional, compact, smooth manifold X are defined by v_k(X)∪x = Sq^k(x) for all x ∈ H^∗(X;Z/2). In particular, v_k = 0 for 2k > n. The Wu classes are connected to the Stiefel-Whitney-classes by the Wu formula (cf. [MS74] p.132)

w_i(X) = X

i+j=k

Sqⁱ(v_j(X)).

Since w₂(T) = 0 =w₂(F) and Hⁱ(T; Z/2) =Hⁱ(F; Z/2) = 0 fori= 1,3 the first three Wu-classes vanish. Consequentlyw4(T) =v4(T) andw6(T) =Sq²(w4(T)).

The even Stiefel-Whitney classes are the mod two reductions of the corresponding Chern classes. We use Section 2.2 to calculatew₄(T) =α₂α₃+α₂α₄+α₃α₄ mod 2. A straight forward calculation shows that this term simplifies to a multiple ofy₁y₂. Thus,w₆(T) = 0 sinceSq²(y₁y₂) = 0.

We already established that g(w₄(T)) = w₄(F), i.e. it is a multiple of g(y₁)g(y₂). In particular, it is a product of classes in degree two. SinceSq²(x) =x² if the degree of x is two, the multiplicative structure of H^∗(F) determines the square Sq²(g(y₁)g(y₂)). It vanishes, whencew₆(F) = 0, too.

This finishes the proof thatg is a polarisation map.

Lemma 8.1 in [CMS10] implies that any ring isomorphism of cohomology Bott mani-folds fixes the Stiefel-Whitney classes. Its proof works along the lines of our explicit calculations to show thatg preserves the Stiefel-Whitney classes.

Remark 4.14. We already mentioned that we can also build the parametric connected sum Fi :=T#_S²Kp, for i= 2,3 and 4, using the embeddings given bys2:=s4◦s3◦i2, s₃:=s₄◦i₃ands₄:=i₄, respectively. The proof of Lemma 4.13 does not use the explicit form of the embedding. Consequently, it also works forF_i.

We are now ready to prove Theorem 4.10.

Proof. First of all we want to understand F, together with appropriate maps, as an element in Ω^String₈ (CP^∞). Let eν_T be a lift of the stable normal Gauss of T.

Then [T, y₁×νe_T] is an element in Ω^String₈ (CP^∞), wherey₁:T →CP^∞is a representative of the homotopy class of maps that corresponds to the generatory₁.

Recall, that the embeddingS² → B4 is the composition s:=s4◦s3◦s2:CP¹₁ → T. In particular, s^∗(y₁) is a generator of S². Thus, S²×D⁶ → T → CP^∞ corresponds to a generator ofH²(S²). Note, that we obtain the other generator by choosing −y₁ instead

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold of y1.

We turn toK_p. The surgery by which we obtainK_p fromK×CP¹ leaves a neighbour-hood of the 2-skeleton invariant. Thus, in a tubular neighbourneighbour-hood S² ×D⁶ ,→ K_p of i: S²,→Kp the mapκ:Kp →CP¹ is the projection to CP¹. The map to CP^∞ is given by composition with the inclusion

e

κ:K_p→CP¹ →CP^∞.

Again precomposing with the embedding S² ×D⁶ → Kp, we obtain a map that also corresponds to a generator of H²(S²).

Thus, the mapsT →CP^∞ and K_p→CP^∞ can be chosen compatibly onS²×D⁶. Since π2(BString) is trivial eν_Kp|S²×D⁶ ' ∗ ' eν_T|S²×D⁶, i.e. the maps T → BString and K_p→BString are also compatible onS²×D⁶.

Hence, the parametric connected sum is a well-defined element [F,(eκ×νe_Kp)#_CP¹(y1×eν_T)

| {z }

:=h×eνF

]∈Ω^String₈ (CP^∞).

Consider S²×D⁶×D¹ together with the String structure given by the constant map pt:S²×D⁶×D¹ →BString and the map S²×D⁶×D¹ → CP^∞ corresponding to a generator ofH²(S²×D⁶×D¹). We obtain a controlled bordism

W :=T ×I∪K_p×I∪S²×D⁶×D¹ betweenF =T#_CP¹K_p and the disjoint union T∪K_p. Thus,

[F, h×νe_F] = [K_p,eκ×νe_Kp] + [T, y₁×eν_T]∈Ω^String₈ (CP^∞).

We prove the Theorem by contradiction.

Assume that there exists a diffeomorphism f:T →F. Then

[T,(h×eν_F)◦f] = [F, h×eν_F] = [K_p,eκ×eν_Kp] + [T, y₁×νe_T]∈Ω^String₈ (CP^∞).

By Lemma 4.12 [Kp,eκ×νeKp] is a non-trivial element in Ω^String₈ (CP^∞). Consequently, it only remains to show that [T,(h×eν_F)◦f] = [T, y₁×νe_T].

By construction the mapsh◦f andy₁toCP^∞correspond to primitive elementsaandy₁ ofH²(T) that square to zero. Thus, there exists an automorphism Ψ :H^∗(T)→H^∗(T) such that Ψ(a) = y1. By assumption all automorphisms on H^∗(T) are realisable by a self-diffeomorphism f⁰. Consequently, there exists a controlled bordismT ×I∪f⁰ T×I between [T,(h×eν_F)◦f] and [T, y₁×eν_T]. Thus, the bordism classes are equal.

Assume F is diffeomorphic to another Bott manifold B₄ by some diffeomorphism ϕ.

Then H^∗(B4) andH^∗(T) are isomorphic by the composition ofϕ^∗ and the polarisation map betweenH^∗(F) andH^∗(T). By assumption B₄ andT are diffeomorphic, implying thatF andT are diffeomorphic which is a contradiction.

4.5 A cohomology Bott manifold which is not diffeomorphic to a Bott manifold Remark 4.15. We can modify this proof such that we see that each cohomology Bott manifoldF_k, fork= 2,3,4, as in Remark 4.14 is not diffeomorphic to any Bott manifold, either. For this purpose we replace [T, y₁×νe_T] in the proof by [T, y_k×νe_T] such that s^∗_k(y_k) is a generator of H²(CP¹_k). With this modifications the proof works the same.

Of course each parametric connected sum,F =:F1as well as eachF_k, induces an element in Ω^String₈ (P3T). Recall that we have a mapP3T →P2T '(CP^∞)⁴. We can compose the mapP3T →(CP^∞)⁴ with the projection to one of the factors. This composition induces four maps

q_k: Ω^String₈ (P3T)→Ω^String₈ (CP^∞).

LetN be a cohomology Bott manifold in the polarised structure set ofT.

Recall the following construction and notation: In Proposition 4.4 we construct a map j3:N → P3T such that the pullback of the generators ai of H²(P3T) are generators of H²(N). The induced element in Ω^String₈ (P3T) is denoted by [N, j₃×eν_N].

The element [T, y_k×eν_T]−[F_k,(j_k)₃×eν_Fk] is non-trivial underq_k. We can use this to show that the [F_k,(j_k)3×eνFk], fork= 1, ...,4 are non-trivial, distinct elements in Ω^String₈ (P3T).

Thus, we obtain representatives for a subgroup of order |Z/2⁴|in Ω^String₈ (P3T).

We conjecture that we can generalise this construction to cohomology Bott manifolds with respect to arbitrary Bott manifolds B₄ which fulfil the (SCRP), i.e. we drop the assumption that the Bott manifold must be String.

On the realisation of some automorphism onH^∗(B4)

5. On the realisation of some automorphism on H

^∗

(B

₄

)

In the previous section we examine cohomology Bott manifolds. Now we return to the original cohomological rigidity problem, to be more precise, the strong cohomological rigidity problem, i.e. the question whether an isomorphism between the cohomology rings of two Bott manifolds can be realised by a diffeomorphism of the underlying spaces.

For Bott manifolds of dimension smaller than or equal to six the strong cohomological rigidity conjecture holds by [Cho11a] and [CM12]. In the latter paper, it is also proven for the so-called Q-trivial Bott manifolds.

In [Cho11a] this question is studied for eight-dimensional Bott manifolds and reduced to the question, whether four automorphisms of the cohomology ring of a special class of Bott manifolds can be realised by a diffeomorphism.

We start by introducing the class of Bott manifolds. Let B₄ be the fourth stage of a Bott tower of the form

CP¹₄ ^{i4 //}B4 π4

=P(γ3⊗γ⁻

1 2A²₃

2 ⊗γ⁻

1 2A¹₃

1 ⊕C) =:P(L3⊕C)

CP¹₃ ^{i3 //}B₃

π3

s4

UU

=P(γ₂^A23 ⊗γ^A₁¹³ ⊕C) =:P(L₂⊕C)

CP¹₂ ^{i2 //}B₂

π2

s3

UU

=P(γ₁^A12⊕C) =:P(L₁⊕C)

CP¹₁

s2

UU

wherec₁(L₁) is arbitrary whilec₁(L₂) =−A²₃y₂−A¹₂y₁ =−α₃ must be divisible by two sincec1(L3) =−α4= ¹₂α3−y3. Here, we use the notation introduced in Section 2.1, i.e.

y_i =−c₁(γ_i), whereγ_i is the tautological bundle over B_i.

For the remainder of the section, B₄ will denote Bott manifolds of this form.

We consider the realisation question for one of the four automorphisms introduced in [Cho11a]. Next, we recall those four automorphisms.

In [Cho11a] these automorphism are defined using the bundle basis, i.e. using the basis consisting of y_i = −c₁(γ_i), for 1 ≤ i ≤ 4. But to attack the realisation question we must understand the automorphism on the basis elements xi, for 1 ≤ i ≤ 4, of the geometric basis introduced in Section 2.1. Recall that the geometric basis is given by the Kronecker duals of the homology classes defined by the CP¹_i, embedded along the appropriate compositions of inclusions of the fibres and sections.

On the realisation of some automorphism onH^∗(B4)

By Proposition 2.3 we get the following base changes between the geometric and the bundle basis for B₄:

x1 =y1, y1 =x1 (8)

x₂ =y₂−α₂, y₂ =x₂+A¹₂x₁

x₃ =y₃−α₃, y₃ =x₃+A²₃x₂+ (A¹₂A²₃+A¹₃)x₁ x₄ =y₄−α₄, y₄ =x₄+x₃+1

2A²₃x₂+1

2(A¹₂A²₃+A¹₃)x₁.

We abbreviateA²₃x₂+ (A¹₂A²₃+A¹₃)x₁ byαe₃. This notation is justified sincex²_i =−eα_ix_i. For the sake of completeness we now recall all four automorphismsφ_i ofH^∗(B₄), defined in [Cho11a] fori= 1,2,3,4, in the bundle basis and in the geometric basis even though we only examineφ₁ later on.

For the two bases the automorphisms are defined by φ_i(y_j) = y_j and φ_i(x_j) = x_j for j= 1,2 and by:

φ₁(y₃) =2y₄−y₃+α₃, φ₁(x₃) =2x₄+x₃ (9) φ₁(y₄) =y₄, φ₁(x₄) =−x₄

φ2(y3) =2y4−y3+α3, φ2(x3) =2x4+x3

φ2(y4) =y4−y3+α₃

2 , φ2(x4) =−x4−x3− αe₃ 2 φ₃(y₃) =−2y₄+y₃, φ₃(x₃) =−2x₄−x₃−αe₃ φ₃(y₄) =−y₄, φ₃(x₄) =x₄

φ4(y3) =−2y4+y3, φ4(x3) =−2x4−x3−αe3

φ4(y4) =−y4+y3−α₃

2 , φ4(x4) =x4+x3+ αe₃ 2 .

We consider the first automorphism φ₁. Its easy form in the geometric basis allows us to apply Corollary 3.12.

Recall that, to apply Corollary 3.12, we need to decompose the manifold on which we want to realise a diffeomorphism. Let B₄ =: M ∪h∂ N, where h_∂: ∂M → ∂N is a diffeomorphism. We determine the explicit forms of M and N later in this section. In particular, we see thatH²(N)∼=H²(B₄). Thus, we can attempt to realiseφ₁onN. This is actually possible by a diffeomorphism n:N → N which we also construct. Finally, we use Corollary 3.12 to examine whether the diffeomorphism can be extended overM.

On the realisation of some automorphism onH^∗(B4)

Recall that Ω^String₈ (pt) ∼= Z⊕Z/2 (cf. [Gia71]), where the two-torsion is generated by an elementθe₈.

Lemma 5.1. The generator θe8 of Z/2 ⊂Ω^String₈ (pt) is the exotic eight-sphere Θ8 con-sidered as an element in String-bordism.

Proof. Since H^k(Θ₈) vanishes for k 6= 0,8 there is no obstruction to the existence of a String structureϑ₈ on Θ₈. Thus, [Θ₈, ϑ₈] =:θe₈ clearly is an element in Ω^String₈ (pt).

It remains to show, thatθe₈ is non-trivial and of order two.

Assume thatθe₈vanishes in Ω^String₈ (pt). Then, there exists a bordismW, together with a String-structureν₈:W →BStringsuch that∂W = Θ₈ andν₈|Θ8 =ϑ₈. This will result in a contradiction to the non-existence of a parallelisable manifold whose boundary is Θ₈.

By surgery below the middle dimension in the interior ofW we can turnν₈ into a four-equivalence. Thus, we can assumeW to be three-connected which implies H⁸(W) = 0.

The obstruction for the existence of a lift of ν₈ to BOh9i → BOh8i = BString is an element inH⁸(W). Hence, we know thatν₈ admits a liftν₉:W →BOh9i.

The obstruction to the existence of a liftν₁₀:BOh9i →BOh10i isw₉(W).

Recall that the Wu classes v_i(X) of an n−dimensional, compact, smooth manifold X are defined by v_k(X)∪x = Sq^k(x) for all x ∈ H^∗(X;Z/2). In particular, v_k = 0 for 2k > n. The Wu classes are connected to the Stiefel-Whitney-classes by the Wu formula (cf. [MS74] p.132)

wi(X) = X

i+j=k

Sqⁱ(vj(X)).

Since W is three-connected the formula for w₉(W) simplifies to

w₉(W) = Sq⁰(v₉(W)) +Sq⁴(v₅(W)) +Sq⁵(v₄(W)) +Sq⁹(v₀(W))

= Sq⁰(v₉(W)) +Sq⁴(v₅(W)),

where the second equality holds for dimension reasons. Butv5(W) andv9(W) correspond toSqⁱ:H^k(W;Z/2)→H^k+i(W;Z/2) fori= 5,9, respectively. Consequently, they also vanish.

This implies that ν10 exists. Since W is of dimension nine all further obstruction toW being parallelisable vanish. But since bP₉ is trivial Θ₈ cannot bound a parallelisable manifold and we have a contradiction. Hence,θe₈ is non-trivial in Ω^String₈ (pt).

Since Θ8#Θ8=S⁸ it is of order two.

Let X ×BString ^E×p−→⁸ BO×BO →^⊕ BO denote a twisted fibration over BO. The inclusionpt ,→X induces a map Ω^String₈ (pt)→ Ω^String₈ (X, E). Let θ₈ denote the image of eθ₈ under this map.

On the realisation of some automorphism onH^∗(B4) Our goal is to prove the following theorem.

Theorem 5.2. Let φ₁:H^∗(B₄)→H^∗(B₄) be the automorphism of Equation (9) ande⁸ an eight-cell. Then there exist

• a twisted fibration

B:= (CP²]CP²∪e⁸)×BString ^{E×p8 //}BO×BO ^{⊕ //}BO ,

• a decompositionB4=M∪h∂N into manifolds with boundary and a diffeomorphism n:N →N,

• two normal three-smoothingseν₁,νe₂:M →Bfulfillingνe₁◦h⁻¹_∂ ◦n◦h_∂ 'eν₂|∂M which give rise to an element [M∪_h⁻¹

∂ ◦n◦h∂M,νe₁∪νe₂] =:ω∈Ω^String₈ (CP²]CP²∪e⁸, E) and

• invariantsa₁, a₂ : Ω^String₈ (CP²]CP²∪e⁸, E)→Z/2 such that φ₁ is realisable if a₁(ω) = 0 =a₂(ω) and ω6=θ₈.

All objects will be constructed in a very explicit way subsequently.

The invariantsa₁ and a₂ are so-called codimension two Arf-invariants. We will see that they allow a very nice geometric description. Roughly, they associate to elements in the torsion subgroup of Ω^String₈ (CP²]CP²∪e⁸, E) the Arf-invariant of some codimension two submanifold.

The second cohomology of N turns out to be isomorphic to H²(B₄). The conjugation of n^∗ with this isomorphism realisesφ₁ on a subspace ofB₄.

To attack the realisation problem for the automorphisms φi for i= 2,3,4 in Equation (9) using Corollary 3.12 is more difficult, if at all possible, since there is no obvious decomposition of B₄ into manifolds M⁰ and N⁰ such that φ_i can be realised on N⁰ in some way.

The proof takes the remainder of the section. It consists of two parts. First, we construct the objects whose existence is claimed in the theorem. Then we use modified surgery theory, in particular Corollary 3.12, to examine if we can extend the self-diffeomorphism, which we can construct on a subspace ofB₄, all over B₄. The subsequent sections can be summarised as follows:

In Section 5.1 we constructM,N and the diffeomorphismh_∂:∂M →∂N. In Section 5.2 we construct the diffeomorphism n:N → N. In Section 5.3 we construct the fibration B and the normal smoothingsνe₁,eν₂:N →B. This finishes the constructive part of the proof.

Next, we compute the twisted bordism group Ω^String₈ (CP²]CP²∪e⁸, E) in Section 5.4.

In Section 5.5 we assemble all objects into a proof of Theorem 5.2. The key of the proof is to develop the invariants a₁ and a₂.

5.1 A suitable description forB4

5.1. A suitable description for B₄

First of all, we change our perspective on the Bott manifold B4 slightly. So far we con-sidered B₄ as CP¹-fibre bundle overB₃. We change that now.

For this purpose, we use Ehresmann’s theorem (cf. [Voi07] Chapter 9.9.1). Let B, E_i and Fi fori= 1,2 be smooth manifolds, p1: E1 → B a smooth fibre bundle with fibre F₁ and p₂:E₂ → E₁ a smooth fibre bundle with fibre F₂. Then Ehresmann’s theorem states that the composition p := p₁ ◦p₂ : E₂ → B is again a smooth fibre bundle if p⁻¹(pt) is compact. In particular, the fibre of p:E₂ → B is E₂|F1, i.e. it is the total space of a fibre bundle F₂ →F →F₁.

In our situation we consider the bundles π₄:B₄ → B₃ and π₃:B₃ → B₂. By Ehres-mann’s Theorem π:=π₃◦π₄ is again a fibre bundle ifπ⁻¹(pt) is compact.

Let i3: CP¹₃ → B3 denote the inclusion of the fibre. To determine what the restriction of B₄ = P(L₃ ⊕C) to i₃(CP¹₃) is, we determine the restriction of the defining bundle L₃, i.e. we consider the pullbacki^∗₃L₃. By definition of the tautological line bundle over some stage Bj, this is just the tautological line bundleγ =i^∗₃(γ3) over CP¹₃. Thus,

π⁻¹(pt) =π₄⁻¹(i₃(CP¹₃)) =P(γ⊕C).

In particular, π⁻¹(pt) is compact. Hence, π:B₄ → B₂ is a fibre bundle with fibre P(γ⊕C).

We have another description for P(γ⊕C).

In [Hir51] Hirzebruch already showed that two Bott manifolds H := P(γ₁^a⊕C) and H⁰ := P(γ₁^a0 ⊕C) of dimension four are diffeomorphic if and only if a= a⁰ mod 2. If a= 0 mod 2, thenHis diffeomorphic toCP¹×CP¹ and we denote it byH0. Otherwise, i.e. if a = 1 mod 2, then H is diffeomorphic to CP²]CP² which we denote by H1. Honoring his work we still call Bott manifolds of dimension fourHirzebruch surfaces.

In Section 2.2 we determine the Stiefel-Whitney classes of a Bott manifold. These results show thatw2(P(γ ⊕C))6= 0. Consequently, the fibre of the bundle π:B4→B2

is diffeomorphic to the non-trivial Hirzebruch surface CP²]CP². Note that the base space B₂ is a Hirzebruch surface, too. But sinceA¹₂ is arbitrary, we do not know which one, so we stick to the notation B2.

The next step is to use the description ofB4 as total space of H1 →B4 →B2 to obtain the manifoldsM and N of Theorem 5.2.

The idea to obtainM and N is the following:

We decompose the base spaceB₂into two parts. One part is the a tubular neighbourhood of the two-skeletonS²∨S²ofB2which we denote byP l, the other part is the complement B₂ −P l, which is the top disc of B₂. We can restrict B₄ to both parts and obtain a decomposition of B₄. Finally, we show that one of this parts admits a diffeomorphism

5.1 A suitable description forB4

toP l× H1.

Now let us construct the decomposition in more detail. Since H1 → B4 → B2 is a locally trivial fibre bundle, the normal bundle of the inclusion of the fibre is trivial.

Thus, there exists an embedding Ψ :D⁴ × H1 ,→ B₄ and we can choose D⁴ such that π◦Ψ(D⁴× H1) =D⁴⊂B2.

We can consider the complementB₄−Ψ(D⁴× H1). This complement is the restriction of the bundleB₄→B₂ toB₂−D⁴.

The Hirzebruch surface B2 is homotopy equivalent to a CW-complex (S² ∨S²)∪e⁴. Thus, if we cut out the top disc, we obtain a space which is homotopy equivalent to S²∨S² ∼=CP¹∨CP¹.

More precisely, we can choose the two-skeleton to consist ofB1=CP¹₁, embedded by the section s₂:CP¹₁ →B₂, and CP¹₂, embedded by the inclusion of the fibre i₂:CP¹₂ →B₂. Their normal bundles areν(CP¹₁ ,→B₂) ∼=γ₁^−A12 and ν(CP¹₂ ,→B₂)∼=C, i.e. the trivial bundle. By the tubular neighbourhood theorem there exist embeddings

s:Dν(CP¹₁)→B₂ and i:Dν(CP¹₂)→B₂.

The images of the embeddings intersect in an embedded D²×D². We identify x₁ ∈ Dν(CP¹₁) andx₂∈Dν(CP¹₂) if s(x₁) =i(x₂) and obtain the plumbing

P l=Dν(CP¹₁)\Dν(CP¹₂)

of Dν(CP¹₁) and Dν(CP¹₂) together with an embedding s\i:P l→B₂.

A priori a plumbing asP l is not a smooth manifold but a manifold with corners. For-tunately we can smoothen the corners by standard methods as described in Appendix A of [Kre10].

We obtain a decompositionD⁴∪(s\i)(P l) =B₂.

If we restrict the composition ofs₄◦s₃ toB₄|(s\i)(P l) we still obtain embeddings of CP¹₁ and CP¹₂ into B₄|(s\i)(P l). If we choose the inclusions of the fibres CP¹₃ and CP¹₄ to be inclusions over a point in (s\i)(P l) and B3|(s\i)(P l), respectively, they are also still em-bedded in B₄|(s\i)(P l). All four embeddings together induce a basis of H₂(B₄|(s\i)(P l)).

Consequently, their Kronecker duals form a basis for H²(B₄|(s\i)(P l)).

Under the inclusionB₄|(s\i)(P l),→B₄ the basis of embeddedCP¹ maps, by definition, to the basis elements σi,1≤i≤4 ofH2(B4) (cf. Section 2.1). The Kronecker duals of the σ_i are the basis elements x_i,1≤i≤4 of the geometric basis. We denote the pullbacks of thex_itoH²(B₄|(s\i)(P l)) by x_i, too. They are Kronecker duals of embeddedCP¹_i, too.

In a sense, we now have a decomposition of B₄ into pieces, namely D⁴ × H1 = B₄|D⁴

andB4|(s\i)(P l)which are identified along the identity on the boundary. Now we want to understandB₄|(s\i)(P l)better, to be able to realiseφ₁ there. Here, realisation means that we find a self-diffeomorphism of B₄|(s\i)(P l) which realises φ₁ on H²(B₄|P l) ∼=H²(B₄).

Im Dokument On the Classification of Cohomology Bott Manifolds (Seite 56-77)